On the Brauer Group of an Arithmetic Model of a Variety over a Global Field of Positive Characteristic
https://doi.org/10.18255/1818-1015-2016-2-164-172
Abstract
Let V be a smooth projective variety over a global field k = κ(C) of rational functions on a smooth projective curve C over a finite field Fq of characteristic p. Assume that there is a projective flat Fq-morphism π : X → C, where X is a smooth projective variety and the generic scheme fiber of π is isomorphic to a variety V (we call π : X → C an arithmetic model of a variety V ).
M. Artin conjectured the finiteness of the Brauer group Br(X) classifying sheaves of Azumaya algebras on X modulo similitude. It is well known that the group Br(X) is contained in the cohomological Brauer group
Br′(X)=H2(X,G ). et m
By definition, the non−p component of the cohomological Brauer group Br′(X) coincides with the direct sum of the l-primary components of the group Br′(X) for all prime numbers l different from the characteristic p. It is known that the structure of k-variety on V yields the canonical morphism of the groups Br(k) → Br′(V ).
The finiteness of the non−p component of the cohomological Brauer group Br′(X) of a variety X has been proved if
[Br′(V )/ Im[Br(k) → Br′(V )]](non −p)
is finite.
In particular, if V is a K 3 surface (in other words, V is a smooth projective simply connected surface
over a field k and the canonical class of a surface of V is trivial: Ω2V = OV ) and the characteristic of the ground field p > 2, then, by the Skorobogatov – Zarhin theorem, [Br′(V )/ Im[Br(k) → Br′(V )]](non −p) is finite, so in this case the groups Br′(X)(non−p) and Br(X)(non−p) are finite.
About the Author
T. V. Prokhorova
A. G. and N. G. Stoletov Vladimir State University, Gorky str., 87, Vladimir, 600000, Russia
Russian Federation
Russian Federation
PhD
References
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For citations:
Prokhorova T.V. On the Brauer Group of an Arithmetic Model of a Variety over a Global Field of Positive Characteristic. Modeling and Analysis of Information Systems. 2016;23(2):164-172. (In Russ.) https://doi.org/10.18255/1818-1015-2016-2-164-172
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